US 11,616,460 B1 · Fuzzy finite-time optimal synchronization control method for fractional-order permanent magnet synchronous generator

Abstract

A fuzzy finite-time optimal synchronization control method for a fractional-order permanent magnet synchronous generator, and belongs to the technical field of generators. A synchronization model between fractional-order driving and driven permanent magnet synchronous generators with capacitance-resistance coupling is established. The dynamic analysis fully reveals that the system has rich dynamic behaviors including chaotic oscillation, and a numerical method provides stability and instability boundaries. Then, under the framework of a fractional-order backstepping control theory, a fuzzy finite-time optimal synchronous control scheme which integrates a hierarchical type-2 fuzzy neural network, a finite-time command filter and a finite-time prescribed performance function is provided.

A fuzzy finite-time optimal synchronization control method for a fractional-order permanent magnet synchronous generator, comprising the following steps:

S1:
modeling a system;
wherein a wind energy conversion system consists of a wind turbine, a permanent magnet synchronous generator and three converters, and the three converters are sequentially a diode bridge rectifier, a DC/DC boost converter and an inverter in the order;
the electric energy generated by the permanent magnet synchronous generator is transmitted to a grid through the converters;
in combination with local aerodynamic characteristics, the power generated by the wind turbine is expressed as:
P
w
=
1
2
⁢
ρπ
⁢
C
p
(
ω
r
⁢
R
/
v
w
,
β
)
⁢
R
2
⁢
v
w
3
(
1
)
wherein ρ, R, ωr, β, ωr and νw represent air density, turbine radius, rotation speed, propeller blade angle, rotation speed and wind speed, respectively, and Cp(ωrR/νw,β) represents turbine power coefficient;
a mechanical fractional-order model of the permanent magnet synchronous generator is provided according to rotation law:
J
⁢
d
α
⁢
ω
r
d
⁢
t
~
α
=
T
s
-
b
⁢
ω
r
-
T
t
(
2
)
wherein α, J, Tt, Tg, {tilde over (t)} and b represent fractional-order coefficient, system inertia, turbine torque, generator torque, time and viscous friction coefficient, respectively;
the torque of an electromagnetic generator is expressed as:
T
g
=
3
2
⁢
p
⁡
(
(
L
q
-
L
d
)
⁢
i
d
⁢
i
q
+
i
q
⁢
ϕ
)
(
3
)
wherein Ld and Lq represent d-axis inductance and q-axis inductance, respectively, id and iq represent d-axis stator current and q-axis stator current, respectively, p and ϕ represent the number of pole pairs and magnetic flux of permanent magnets in a three-phase permanent magnet synchronous generator, respectively;
the fractional-order model of the permanent magnet synchronous generator in a synchronous rotating d-q reference frame is expressed as:
{
d
α
⁢
i
d
d
⁢
t
~
α
=
(
-
R
s
⁢
i
d
+
ω
r
⁢
L
q
⁢
i
q
+
V
d
)
/
L
d
d
α
⁢
i
q
d
⁢
t
~
α
=
(
-
R
s
⁢
i
q
-
(
L
d
⁢
i
d
+
ϕ
)
⁢
ω
r
+
V
q
)
/
L
q
(
4
)
wherein Rs, Vd and Vq represent stator resistance, d-axis stator voltage and q-axis stator current, respectively;
in the Laplace domain, a fractional-order integrator in the form of linear approximation with a zero pole pair is represented by a transfer function with a slope of −20m dB/decade in a Bode diagram:
F
⁡
(
s
)
=
1
s
α
≈
∏
i
=
0
N
f
-
1
(
1
+
s
/
Q
i
)
/
∏
i
=
0
N
f
-
1
(
1
+
s
/
P
i
)
(
5
)
wherein
N
f
=
integer
(
log
⁡
(
ω
max
/
p
f
)
log
⁡
(
1
⁢
0
d
f
/
10
⁢
(
1
-
α
)
·
10
d
f
/
10
⁢
α
)
)
+
1
,
and Pƒ, ωmax and dƒ represent difference values between angular frequency, bandwidth and actual line and approximate line, respectively;
Qi and Pi represent zero and pole of singularity function, respectively;
L=Ld=Lq is obtained due to symmetrical stator windings;
a fractional-order model of the permanent magnet synchronous generator is defined by equation (2) and equation (4) as follows:
{
C
𝒟
0
,
t
α
⁢
ω
r
=
(
3
⁢
p
⁢
ϕ
⁢
i
q
/
2
-
b
⁢
ω
r
-
T
t
)
/
J
,
C
𝒟
0
,
t
α
⁢
i
q
=
(
-
R
s
⁢
i
q
-
(
L
⁢
i
d
+
ϕ
)
⁢
ω
r
+
V
q
)
/
L
,
C
𝒟
0
,
t
α
⁢
i
d
=
(
-
R
s
⁢
i
d
+
ω
r
⁢
L
⁢
i
q
+
V
d
)
/
L
,
(
6
)
wherein C represents Caputo fractional-order derivative with α>0 and starting point at origin;
by introducing new variables x1=Lωr/Rs, x2=pLϕiq/bRs and x3=pLϕid/bRs, a normalized fractional-order model of a permanent magnet synchronous main motor is expressed as:
{
C
𝒟
0
,
t
α
⁢
x
1
=
σ
⁢
x
2
-
ρ
⁢
x
1
+
T
L
C
𝒟
0
,
t
α
⁢
x
2
=
-
x
2
-
x
1
⁢
x
3
+
μ
⁢
x
1
+
u
q
C
𝒟
0
,
t
α
⁢
x
3
=
-
x
3
+
x
1
⁢
x
2
+
u
d
(
7
)
wherein t={tilde over (t)}Rs/L, uq=pLϕVq/bRs2, ud=pLϕVd/bRs2, TL=−L2Tt/JRs2, μ=−pϕ2/bRs, σ=3Lb/2JRs, and ρ=bL/JRs, and x1, x2, x3, t, uq, ud and TL represent normalized angular velocity, q-axis current, d-axis current, time, q-axis voltage, d-axis voltage and load torque, respectively;
σ, ρ and μ represent system parameters;
a fractional-order model of the driven permanent magnet synchronous generator is established by using a Heaviside function H(t−Tg):
{
C
𝒟
0
,
t
α
⁢
y
1
=
σ
⁢
y
2
-
ρ
⁢
y
1
+
T
L
C
𝒟
0
,
t
α
⁢
y
2
=
-
y
2
-
y
1
⁢
y
3
+
μ
⁢
y
1
-
(
κ
1
(
y
1
-
x
1
)
+
κ
2
(
y
2
-
x
2
)
)
·
H
⁡
(
t
-
t
g
)
+
u
q
+
u
2
C
𝒟
0
,
t
α
⁢
y
3
=
-
y
3
+
y
1
⁢
y
2
+
u
d
+
u
3
(
8
)
wherein tg, κ1 and κ2 represent primary synchronous time, the capacitance coupling and the resistance coupling, respectively, and u2 and u3 represent control inputs;
synchronization errors are defined as e1=y1−x1, e2=y2−x2 and e3=y3−x3;
the following equation is obtained by subtracting equation (7) from equation (8):
{
C
𝒟
0
,
t
α
⁢
e
1
=
σ
⁢
e
2
-
ρ
⁢
e
1
,
C
𝒟
0
,
t
α
⁢
e
2
=
-
e
2
-
y
1
⁢
y
3
+
x
1
⁢
x
3
+
μ
⁢
e
1
-
(
κ
1
⁢
e
1
+
κ
2
⁢
e
2
)
⁢
H
⁡
(
t
-
t
g
)
+
u
2
,
C
𝒟
0
,
t
α
⁢
e
3
=
-
e
3
+
y
1
⁢
y
2
-
x
1
⁢
x
2
+
u
3
.
(
9
)
definition 1:
for a sufficiently differentiable function F(t), the Caputo fractional-order derivative is expressed as:
C
𝒟
0
,
t
α
⁢
(
t
)
=
1
Γ
⁡
(
n
-
α
)
⁢
∫
0
t
F
(
n
)
(
τ
)
(
t
-
τ
)
1
-
n
+
α
⁢
d
⁢
τ
(
10
)
wherein
Γ
⁡
(
n
-
α
)
=
∫
0
∞
e
-
t
⁢
t
n
-
α
-
1
⁢
d
⁢
t
represents Euler gamma function, n−1<α<n, and n∈;
a Laplace transform is performed on equation (10) to obtain:
L
⁢
{
C
𝒟
0
,
t
α
⁢
F
⁡
(
t
)
}
=
s
α
⁢
L
⁢
{
F
⁡
(
t
)
}
-
∑
k
=
0
n
-
1
F
(
k
)
(
0
)
s
k
+
1
-
α
(
11
)
for any continuous function, when 0<α<1 and F1(t) and F2(t) are within an interval [0,tζ], the following equation is obtained:
F
1
(
t
)
·
C
𝒟
0
,
t
α
⁢
F
2
(
t
)
+
F
2
(
t
)
·
C
𝒟
0
,
t
α
⁢
F
1
(
t
)
=
C
𝒟
0
,
t
α
⁢
(
F
1
(
t
)
·
F
2
(
t
)
)
+
α
Γ
⁡
(
1
-
α
)
⁢
∫
0
t
1
(
t
-
ς
)
1
-
α
⁢
(
∫
0
ς
F
1
′
(
ϛ
1
)
⁢
d
⁢
ϛ
1
(
t
-
ϛ
1
)
α
·
∫
0
ς
F
2
′
(
ϛ
2
)
⁢
d
⁢
ϛ
2
(
t
-
ϛ
2
)
α
)
⁢
d
⁢
ϛ
(
12
)
when F1(t)=F2(t), the following inequation is derived as:
C(F2(t))≤2F(t)·CF(t) (13)
lamma 1:
for any xs, ys∈, ϑ(xs, ys) is any positive real function and then the following inequation holds if cs>0 and ds>0;
❘
"\[LeftBracketingBar]"
x
s
❘
"\[RightBracketingBar]"
c
s
⁢
❘
"\[LeftBracketingBar]"
y
s
❘
"\[RightBracketingBar]"
d
s
≤
c
s
⁢
ϑ
⁡
(
x
s
,
y
s
)
⁢
❘
"\[LeftBracketingBar]"
x
s
❘
"\[RightBracketingBar]"
c
s
+
d
s
c
s
+
d
s
+
d
s
⁢
ϑ
⁡
(
x
s
,
y
s
)
c
s
d
s
⁢
❘
"\[LeftBracketingBar]"
y
s
❘
"\[RightBracketingBar]"
c
s
+
d
s
c
s
+
d
s
(
14
)
definition 2:
a minimum performance cost function is as follows:
J
¯
=
∫
0
∞
(
Q
¯
(
S
¯
(
t
)
)
+
U
T
⁢
R
¯
⁢
U
)
⁢
dt
(
15
)
wherein S, U and R represent a penalty function, an optimal control input and an N-order matrix, respectively, and Q(S)>0;
S2:
designing a numerical method, applying the numerical method to solve a nonlinear fractional-order system, establishing a fractional-order differential solver comprising a fractional-order value, a system dimension, a start-stop time, a time span, a system initial value, a Gram-Schmidt step size, multilayer correction iteration and convergence error, outputting data, storing the data in a double precision matrix format, and setting the unlimited number of warning information and the minimum step size of machine precision;
and
S3:
establishing a hierarchical type-2 fuzzy neural network and designing a controller that provides an ordered and coordinated motion for the fractional-order permanent magnet synchronous generator.

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Fuzzy finite-time optimal synchronization control method for fractional-order permanent magnet synchronous generator

Life of this patent

Fuzzy finite-time optimal synchronization control method for fractional-order permanent magnet synchronous generator

Life of this patent

Abstract

Claims3